Sequences and Series

In Product of Sums (POS) form, the expression is a product of terms where each term is an OR sum of literals.

In Product of Sums (POS), the overall expression is an AND operation of several OR terms.

( As there are 3 unlike terms)

fact

t - 5tz + 2tz - z + z - t = - 3tz

2: (c) 10th [Soln: an = a + (n - 1)× d where d = -3, Let an = 0⇒ 0 = 24 + (n - 1) × -3⇒ 0 = 24 -3n + 3⇒ 3n = 27⇒ n = 9⇒ 10th term will be negative (-ve).

[(190-15)/7]+1=25+1=26

The GP can be interpreted as 16,8,4,2,1..... . The sum of the GP is a/(1-r)=16/(1-.5)=32

This is an arithmetico-geometric series: S = 2 + 5/2 + 8/4 + 11/8 + ... The sum of an infinite AGP is S = a/(1-r) + dr/(1-r)^2, where a=2, d=3, r=1/2. S = 2/(0.5) + (3*0.5)/(0.5^2) = 4 + 1.5/0.25 = 4 + 6 = 10.

Correct option is (2).

n^th term = dn + (a – d) = 2n + (– 2 – 2) = 2n – 4

n^th term = a + (n – 1)d + ^{${\frac 12}$} (n – 1) (n – 2) c a = 10, d = 4, c = 1, n = 19               = 10 + (19 – 1) 4 + ^{${\frac 12}(19-1)(19-2)$}1              = 10 + 18 ^$\times$ 4 + 9 ^$\times$ 17              = 235

14^th term of the respective A.P. will be calculated as follows: T₁₄ = a₁ + (14 - 1)d 14^th term = 7 + (13) 3 = 46

 a = 3 and d = 5 (18 - 13, 13 - 8, 8 - 3 = 5) 78 = a_n = a + (n - 1)d 78 = 3 + (n - 1)5 78 = 3 + 5n - 5 ^{$\Rightarrow$} 78 = 5n - 2 ^{$\Rightarrow$} 78 + 2 = 5n 80 = 5n ^{$\Rightarrow$} n = 16. So, 78 is the 16^th term of the AP.

7 (a + 6d) = 11 (a + 10d) ^{$\Rightarrow$} 7a + 42d = 11a + 110d 7a - 11a = 110d - 42d ^{$\Rightarrow$} -4a = 68d -a = 17d ^{$\Rightarrow$} a = -17d Hence, a₁₈ = a + 17d  a₁₈ = -17d + 17d = 0.